The problem of generating frames by iterative actions of operators [1–3] has emerged within the research related to the dynamical sampling problem [1, 3–6]. The conditions under which a frame generated by iterative actions of operators exists for a finite-dimensional or a separable Hilbert space have been stated in Aldroubi et al. [1, 3]. If we have a frame, then a linear combination of a dual frame with the dynamically sampled coefficients reproduce the original signal. The natural follow-up questions to ask in this setup are: whether we can obtain a scalable [7–13] frame under iterative actions, and if not, whether we can find a dual frame which preserves the dynamical structure.

Let A be an operator on a separable Hilbert space ℍ. We consider a countable set of vectors G in ℍ, and a function L:G → ℕ₀, where ℕ₀ = ℕ∪{0}. Related to the iterated system of vectors

we answer the following two questions:

The motivation for studying systems of type (Equation 1) comes from the dynamical sampling problem (DSP): Let the initial state of a dynamical system be represented by an unknown element f ∈ ℍ. Say the initial state f is evolving under the action of an operator A^* to the states fj=A*fj-1, where f₀ = f and j∈ℕ0+. Given a set of vectors G ⊂ ℍ, one can find conditions on A, G and L = L(g) which allow the recovery of the initial state f from the set of samples {〈A*jf,g〉|g∈G}j=0L(g). In short, the problem of signal recovery via dynamical sampling is solvable if the set of vectors (Equation 1) is a frame for ℍ, [1, 3]. In frame theory it is known that every frame has at least one dual frame; if FAL(G) is a frame for ℍ, and its dual frame elements are h_{g, j}, then all f ∈ ℍ are reconstructed as

Throughout this paper ℍ denotes a separable Hilbert space. Given an index set I, a sequence F = {f_i}_i∈I of nonzero elements of ℍ is a frame for ℍ, if there exist 0 < A ≤ B < ∞ such that

In finite dimensions, we find it useful to express frames as matrices, so we abuse the notation of F as follows: when dim ℍ = n, a frame F = {f_i}_i∈I for ℍ is often represented by a n × k matrix F, whose column vectors are f_i, 1 ≤ i ≤ k (k≥n). The frame operator S = FF^* is then positive, self-adjoint and invertible.

For each frame F there exists at least one dual frame G = {g_i}_i∈I, satisfying

The matrix equation FG^* = GF^* = I is an equivalent expression to the frame representation (Equation 4). The set {gi=S-1fi}i∈I is called the canonical dual frame.

Finding a dual frame can be computationally challenging; thus it is of interest to work with tight or scalable frames. We say that a frame is A-tight if A = B in Equation (3). When A = 1, we call F a Parseval frame. If a frame F = {f_i}_i∈I is not tight, but we can find scaling coefficients w_i ≥ 0, i ∈ I, such that the scaled frame F_w = {w_if_i}_i∈I is tight, then we call the original frame F a scalable frame. We note that the notion of scalability of a frame is defined for a unit-norm frame in [10], but in this manuscript we do not require a scalable frame to be unit-norm. If the scaling coefficients w_i are positive for all i ∈ I, then we call the original frame F a strictly scalable frame.

Let I denote a finite or countable index set, let G = {_{fs}s ∈ I} ⊂ ℍ and let A:ℍ → ℍ be a bounded operator. We call the collection

a dynamical system, where L_s ≥ 0 (L_s may go to ∞) and L = (L_s)_s∈I is a sequence of iterations related to the set G. The operator A, involved in generating the set Equation (5), is referred to as a dynamical operator. If A is fixed, then we use the notation FGL, and if G = {f} and L = {L}, then we label (Equation 5) by FfL.

Note that in Aldroubi et al. [1], f_s are chosen to be the standard basis vectors, while in this manuscript, we allow the use of any nonzero vector f_s ∈ ℍ. If Equation (5) is a frame for ℍ, then we call FGL(A) a dynamical frame, generated by a dynamical operator A, set G and sequence of iterations L.

Theorem 1. Let G = {f_s}_s∈I ⊂ ℍ, where I is a countable index set, and assume that FGL(A) is a frame for ℍ, with frame operator S. The canonical dual frame of FGL(A) is the dynamical frame FG′L(B), generated by operator B = S⁻¹AS, G′={gs=S-1fs}s∈I, and sequence of iterations L.

In addition, every f ∈ ℍ can be reconstructed from the dynamical samples {〈A*jf,fs〉|0≤j≤Ls}s∈I via the frame reconstruction formula

Proof: Let FGL(A) be a frame for ℍ, where G = {f_s}_s∈I ⊂ ℍ and I is countable, and assume that S is the frame operator for FGL(A). Then S is an invertible operator, and for any Ajfs∈FGL(A) we have that S-1(Ajfs) is a canonical dual frame element, where s ∈ I, 0 ≤ j ≤ L_s. If we choose

then for all j ≥ 0 we have

     □

Theorem 2. Let ℍ₁ and ℍ₂ be two separable Hilbert spaces. Let A be a bounded operator on ℍ₁ and let B:ℍ₁ → ℍ₂ be an invertible operator. Let I be a countable index set, and fix G = {f_s}_s∈I ⊂ ℍ₁. Set g_s = Bf_s ∈ ℍ₂, s ∈ I, and C: = BAB⁻¹. For any set of iterations L = (L_s)_s∈I, L_s ≥ 0, TFAE:

Proof: Let g_s = Bf_s ∈ ℍ₂, s ∈ I, and set C = BAB⁻¹. Note that C^j = BA^jB⁻¹, due to B⁻¹B = I. For any Ajfs∈F⊂ℍ1, we have

The operator B is invertible, thus BF is a frame if and only if F is a frame, so (i) and (ii) are equivalent.     □

If the operator B occurring in Theorem 2 is unitary and ℍ₁ = ℍ₂, then the property of scalability is preserved. We have:

Corollary 1. Let f_s ∈ ℍ, and set vs=U*fs for all s ∈ I, where I is a countable index set. Let A, R be two operators on a separable Hilbert space ℍ, and let U be a unitary operator on ℍ. If A = URU^*, then TFAE:

In the next section, we exploit the simplicity of the unitary diagonalization of normal operators to give more explicit conditions on the normal operator A in order to ensure scalability of a frame of type FGL(A).

In this section, we explore the case when the operator A is of block-diagonal form. Block-diagonal operators give us a chance to offer a partial answer to (Q1) in the case when we don't have a unitary diagonalization.

Let A_s:ℍ_s → ℍ_s be an operator on ℍ_s, with dim ℍ_s = n_s, 1 ≤ s ≤ p. Let A:ℍ → ℍ be a block-diagonal operator on ℍ=⊕s=1pℍs, constructed as follows:

Let v ∈ ℍ_s for some 1 ≤ s ≤ p. If ℍ is the direct sum of a family (ℍ_s)_s of Hilbert space, for each index s there exists a canonical inclusion i_s of ℍ_s into H. We say that v is well-embeded in f ∈ ℍ with respect to operator (Equation 12) if f = i_s(v).

Theorem 5. Let A_s:ℍ_s → ℍ_s be an operator on ℍ_s, with dim ℍ_s = n_s, 1 ≤ s ≤ p. Let A:ℍ → ℍ be a block-diagonal operator on ℍ=⊕s=1pℍs, constructed as in Equation (12). Let v_s,1…, v_s,ms ∈ ℍ_s, 1 ≤ s ≤ p be well-embedded in f_s,1, …, f_s,ms ∈ ℍ, 1 ≤ s ≤ p.

is a (scalable) frame of ℍ if and only if {Asjvs,k|1≤k≤ms}j=0Ls,k are (scalable) frames of ℍ_s for all 1 ≤ s ≤ p.

Proof: We assume that all m_s = 1, i.e., f_s,k = f_s, v_s,k = v_s, and L_s,k = L_s, 1 ≤ s ≤ p, to simplify the presentation of the proof. The matrix representation of ∪s=1p{Ajfs}j=0Ls with scaling coefficients w_{s, j}, 0 ≤ j ≤ L_s for each s = 1, …, p is of block-diagonal form:

If F is a tight frame, then row vectors of F are orthogonal and have the same norm and so does (ws,0vs…ws,LsAsLsvs) for each s = 1, …, p. This implies that the system {Asjvs}j=0Lk is a scalable frame for ℍ_s for all 1 ≤ s ≤ p.

Now, suppose that for each 1 ≤ s ≤ p, the system {Ajvs}j=0Ls is a scalable frame for ℍ_s. Then, there exist some scaling coefficients w_{s, j}, 1 ≤ s ≤ p, 0 ≤ j ≤ L_s, such that {ws,jAsjvs|0≤j≤Ls} is a Parseval frame for each s = 1, …p.     □

Block operators we can employ in Theorem 5 include:

Example 1. Let a, b, c, d be real numbers.

The conditions for Example 1(b) are obtained from the following proposition:

Proposition 1. Let a, b, c, d be real numbers such that a > 0 and abcd ≠ 0. Then the following two statements are equivalent:

is a strictly scalable frame for ℝ².

Proof. We first note that the condition 0<-acbd<1 is equivalent to (a>0,-bc>ad>0) or (a>0,-da>cb>0).

(1) ⇒ (2): The conditions a>0,-bc>ad>0 imply that

and the conditions a>0,-da>cb>0 imply that

Then

are positive numbers and

is a Parseval frame for ℝ².

(1) ⇐ (2): If the system F is strictly scalable, then the normalized system

is a unit-norm scalable frame. The Gramian matrix of diagram vectors of F′ has positive scalings in its null space:

Inequality (Equation 15) implies that -acbd>0. Next we show that -acbd<1.

In case b > 0, inequality (Equation 14) implies that

If (c > 0 and ac + bd ≥ 0) or (c < 0 and ac + bd ≤ 0), then a²cd + abd² > 0, which implies -acbd<1. If c > 0 and ac + bd < 0, then ac < −bd, which implies 1<-bdac since ac > 0. Similarly, if c < 0 and ac + bd > 0, then ac > −bd, which implies 1<-bdac since ac < 0. This is equivalent to -acbd<1.

In case b < 0, suppose that -acbd≥1. Multiply both sides by the positive number −abd². On one hand we have a²cd ≥ −abd² and on the other hand, from inequality (14), we have a²cd − abc² < −abd² + b²cd. Since a²cd ≥ −abd², we have −abd² − abc² < −abd² + b²cd, which implies -acbd<1. This contradicts our assumption.     □

Proposition 2. Let i, j, k, l ∈ ℕ be such that p < k ≤ n, q < l ≤ n, and let N ∈ ℕ. For each m = 1, …, N, we define Aklpq(m)=[aij(m)]i,j=1n as

If for each m = 1, …, N, a_m, b_m, c_m and d_m satisfy the conditions of Proposition 1, and the system

spans ℝⁿ, then Equation (16) is a strictly scalable frame for ℝⁿ.

Let a₁, …, a_n ∈ ℝ which are not all zeros; then

is called a companion operator [14].

Proposition 3. Let the dynamical operator A be a companion operator (18) in ℝⁿ, then we have

It is known that the standard orthonormal basis B can not be extended to a scalable frame by adding one vector f ∈ ℍ\B, [8, 11]. Thus, we explore when one can generate a dynamical frame by adding two vectors. Although a companion operator A does not generate a scalable frame Fe1n, it can generate a scalable frame Fe1n+1 under certain conditions. Using the companion operator A, we have

where

Proposition 4. [11] Let {e₁, …e_n} be the standard orthonormal basis in ℝⁿ with n ≥ 2. Let f and g be two unit-norm vectors in ℝⁿ.

If either system {e₁, …e_n, f, g} or {e₁, …e_n−1, f, g} is scalable, then f and g have only two nonzero elements in the same entries.

We now assume that Fe1n+1 is scalable. Then by Proposition 4, a_m = 0 implies that a_m−1 = 0 for m ≥ 2. This implies that a₁ = … = a_n−2 = 0. A consequence of this is the following result.

Proposition 5. Let a and b be real numbers such that a > 0 and 0<-a2a+b2<1. Then the companion operator A in ℝⁿ,

generates a strictly scalable frame Fe1n+1.

Proof: We have

which is strictly scalable.     □

We note that the operator A in Equation (20) is not diagonalizable.

Example 2. Let a and b be real numbers such that 0<-a(a2+bc)b2(a+d)<1 and a > 0. Then the operator

generates a strictly scalable frame Fe1n for ℝⁿ.

Proof: We have

The strict scalability follows by Example 1.     □

Example 3. Let 2 sin²(ϕ) − 1 > 0 and let

The set

is a strictly scalable frame of ℝⁿ.

We have studied the scalability of dynamical frames in a separable Hilbert space ℍ. Given an operator A on ℍ and a (at most countable) set G ⊂ ℍ, we have explored the relations between A, G and the number of iterations that make the system (1) a scalable frame. In section 3.1 we have fully answered question (Q1) in finite dimensions, when working with a normal operator. We have also studied operators with specialized structures, such as block-diagonal operators (section 4), and companion operators (section 5), which are not necessarily normal; for instance, note that the block-diagonal matrix A in Theorem 5 cannot be normal if one of its blocks is not normal.

In adition, we have established the canonical dual frame for frames of type F_G(A); in particular, we showed that the canonical dual frame has, as anticipated, an iterative set structure; in section 4.1, we had generalized the notion of dynamical frames to the notion of frames generated by sequences of operators, and verified that the canonical dual frame result is generalized as well (Theorem 6); this type of result holds true in any separable Hilbert space ℍ.

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.